Dyck path triangulations and extendability
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1 Dyck path triangulations and extendability [extended abstract] Cesar Ceballos 1,2 Arnau Padrol 3 Camilo Sarmiento 4 1 York University 2 Fields Institute 3 Freie Universität Berlin 4 Otto-von-Guericke-Universität Magdeburg 27th FPSAC 2015, Daejeon July 10th, 2015 arxiv: / 17
2 Outline of the talk Introduction to triangulations of d 1 n 1 Dyck path triangulations and some relatives Extendability of partial triangulations Further questions 0 / 17
3 Outline of the talk Introduction to triangulations of d 1 n 1 Dyck path triangulations and some relatives Extendability of partial triangulations Further questions 0 / 17
4 Objects of study The standard (d 1)-simplex is the polytope: d 1 := conv {e 1, e 2,..., e d } R d 1 / 17
5 Objects of study The product of simplices is the (d + n 2)-dimensional polytope: { d 1 n 1 := conv (e i, e j ): e i R d, e j R n} R d+n 2 / 17
6 Objects of study A triangulation of d 1 n 1 is a subdivision into simplices (spanned by vertices of d 1 n 1 ) that cover d 1 n 1 and intersect properly: 3 / 17
7 Motivation: connection to other areas Subdivisions of d 1 n 1 appear in many different contexts: Geometric combinatorics [Haiman, Santos] Toric geometry and algebra [Sturmfels-Zelevinsky, Babson-Billera] Tropical geometry [Sturmfels-Develin, Santos, Kapranov, Speyer, Joswig-Herrmann-Speyer, Fink-Rincon] Matroid of lines in generic arrangement of flags [Ardila-Billey] 4 / 17
8 Some results available It is an open problem due to Gel fand, Kapranov and Zelevinsky to find an explicit description of all triangulations of d 1 n 1. [Sturmfels 91] Number of triangulations known only for d 1 1 and (asymptotically in d) for d 1 2 [Santos] Connectedness of graphs of triangulations known only for d 1 1 and of d 1 2 are connected under flips [Santos] Secondary polytope of d 1 1 is affinely isomorphic to (d 1)-permutahedron [Gelfand-Kapranov-Zelevinsky] Some facets of secondary polytope of d 1 n 1 have arbitrarily large integral normal vectors [Babson-Billera] 5 / 17
9 Preliminaries: grid representation 6 / 17
10 Preliminaries: grid representation View a simplex of d 1 n 1 as a subset of a d n grid: 6 / 17
11 Preliminaries: grid representation View a simplex of d 1 n 1 as a subset of a d n grid: So a triangulation of d 1 n 1 looks like: 6 / 17
12 Preliminaries: staircase triangulation A staircase in the d n grid represents a full-dimensional simplex of d 1 n 1 7 / 17
13 Preliminaries: staircase triangulation A staircase in the d n grid represents a full-dimensional simplex of d 1 n 1 The ( ) d+n 2 d 1 staircases represent simplices that form the staircase triangulation of d 1 n 1 : 7 / 17
14 Preliminaries: mixed subdivisions 8 / 17
15 Preliminaries: mixed subdivisions View columns in grid representation as labelled Minkowski summands: 8 / 17
16 Preliminaries: mixed subdivisions View columns in grid representation as labelled Minkowski summands: 8 / 17
17 Preliminaries: mixed subdivisions View columns in grid representation as labelled Minkowski summands: Theorem (Sturmfels 94, Huber-Rambau-Santos 00) { } { triangulations fine mixed subdivisions of of d 1 n 1 Cayley trick d n 1 := n n 1 } 8 / 17
18 Preliminaries: mixed subdivisions Theorem (Sturmfels 94, Huber-Rambau-Santos 00) { } { } triangulations Cayley trick fine mixed subdivisions of of d 1 n 1 d n 1 := n n 1 Examples Triangulation of 1 2 fine mixed subdivision of / 17
19 Preliminaries: mixed subdivisions Theorem (Sturmfels 94, Huber-Rambau-Santos 00) { } { } triangulations Cayley trick fine mixed subdivisions of of d 1 n 1 d n 1 := n n 1 Examples Triangulation of 3 2 fine mixed subdivision of / 17
20 Outline of the talk Introduction to triangulations of d 1 n 1 Dyck path triangulations and some relatives Extendability of partial triangulations Further questions 9 / 17
21 Dyck path triangulation of n 1 n 1 Consider the Dyck paths in an n n grid, 10 / 17
22 Dyck path triangulation of n 1 n 1 Consider the Dyck paths in an n n grid, with their orbits under diagonal cyclic action (i, j) (i + 1 mod n, j + 1 mod n) 10 / 17
23 Dyck path triangulation of n 1 n 1 Consider the Dyck paths in an n n grid, with their orbits under diagonal cyclic action (i, j) (i + 1 mod n, j + 1 mod n) 10 / 17
24 Dyck path triangulation of n 1 n 1 Consider the Dyck paths in an n n grid, with their orbits under diagonal cyclic action (i, j) (i + 1 mod n, j + 1 mod n) Theorem (Ceballos-Padrol-S 14) The resulting n 1 ( 2(n 1) ) n n 1 simplices form a (regular) triangulation of n 1 n 1 : the Dyck path triangulation. 10 / 17
25 Dyck path triangulation of n 1 n 1 Consider the Dyck paths in an n n grid, with their orbits under diagonal cyclic action (i, j) (i + 1 mod n, j + 1 mod n) Theorem (Ceballos-Padrol-S 14) The resulting n 1 ( 2(n 1) ) n n 1 simplices form a (regular) triangulation of n 1 n 1 : the Dyck path triangulation. 10 / 17
26 Some relatives 11 / 17
27 Some relatives Theorem (Ceballos-Padrol-S 14) The following are (regular) triangulations. 11 / 17
28 Some relatives Flipped Dyck path triangulation: Theorem (Ceballos-Padrol-S 14) The following are (regular) triangulations. 11 / 17
29 Some relatives Flipped Dyck path triangulation: Theorem (Ceballos-Padrol-S 14) The following are (regular) triangulations. 11 / 17
30 Some relatives Flipped Dyck path triangulation: Theorem (Ceballos-Padrol-S 14) The following are (regular) triangulations. Extended Dyck path triangulation: 11 / 17
31 Some relatives Flipped Dyck path triangulation: Theorem (Ceballos-Padrol-S 14) The following are (regular) triangulations. Extended Dyck path triangulation: 11 / 17
32 Some relatives Flipped Dyck path triangulation: Theorem (Ceballos-Padrol-S 14) The following are (regular) triangulations. Extended Dyck path triangulation: Rational Dyck path triangulation 11 / 17
33 Some relatives Flipped Dyck path triangulation: Theorem (Ceballos-Padrol-S 14) The following are (regular) triangulations. Extended Dyck path triangulation: Rational Dyck path triangulation 11 / 17
34 Some relatives Flipped Dyck path triangulation: Theorem (Ceballos-Padrol-S 14) The following are (regular) triangulations. Extended Dyck path triangulation: Rational Dyck path triangulation 11 / 17
35 Outline of the talk Introduction to triangulations of d 1 n 1 Dyck path triangulations and some relatives Extendability of partial triangulations Further questions 11 / 17
36 Partial triangulations Partial triangulation = triangulation of skel k 1 ( d 1 ) n 1 12 / 17
37 Partial triangulations Partial triangulation = triangulation of skel k 1 ( d 1 ) n 1 12 / 17
38 Partial triangulations Partial triangulation = triangulation of skel k 1 ( d 1 ) n 1 12 / 17
39 Partial triangulations Partial triangulation = triangulation of skel k 1 ( d 1 ) n 1 12 / 17
40 Partial triangulations Partial triangulation = triangulation of skel k 1 ( d 1 ) n 1 12 / 17
41 Partial triangulations Partial triangulation = triangulation of skel k 1 ( d 1 ) n 1 12 / 17
42 Partial triangulations Partial triangulation = triangulation of skel k 1 ( d 1 ) n 1 Question: are there d, n, k N + conditions such that every triangulation of skel k 1 ( d 1 ) n 1 extends? 12 / 17
43 Triangulations of skel k 1 ( d 1 ) n 1 Motivation and existing results 13 / 17
44 Triangulations of skel k 1 ( d 1 ) n 1 Motivation and existing results k = 2, min{d, n} 3: one obstruction, complete characterization [Ardila-Ceballos 11] 13 / 17
45 Triangulations of skel k 1 ( d 1 ) n 1 Motivation and existing results k = 2, min{d, n} 3: one obstruction, complete characterization [Ardila-Ceballos 11] k = 2, min{d, n} > 3: more obstructions, open [Santos 11, Ceballos-Padrol-S. 14] 13 / 17
46 Triangulations of skel k 1 ( d 1 ) n 1 Motivation and existing results k = 2, min{d, n} 3: one obstruction, complete characterization [Ardila-Ceballos 11] k = 2, min{d, n} > 3: more obstructions, open [Santos 11, Ceballos-Padrol-S. 14] Conjecture: for k = 2, general d, n, there are -many obstructions 13 / 17
47 Triangulations of skel k 1 ( d 1 ) n 1 Motivation and existing results k = 2, min{d, n} 3: one obstruction, complete characterization [Ardila-Ceballos 11] k = 2, min{d, n} > 3: more obstructions, open [Santos 11, Ceballos-Padrol-S. 14] Conjecture: for k = 2, general d, n, there are -many obstructions d n > k: open. Conjecture: there are -many obstructions 13 / 17
48 Triangulations of skel k 1 ( d 1 ) n 1 Motivation and existing results k = 2, min{d, n} 3: one obstruction, complete characterization [Ardila-Ceballos 11] k = 2, min{d, n} > 3: more obstructions, open [Santos 11, Ceballos-Padrol-S. 14] Conjecture: for k = 2, general d, n, there are -many obstructions d n > k: open. Conjecture: there are -many obstructions d k n: solved [Ceballos-Padrol-S. 14]. 13 / 17
49 Extendability result Theorem (Ceballos-Padrol-S 14) Let d k > n N. Every triangulation of skel k 1 ( d 1 ) n 1 extends to a unique triangulation of d 1 n / 17
50 Extendability result Theorem (Ceballos-Padrol-S 14) Let d k > n N. Every triangulation of skel k 1 ( d 1 ) n 1 extends to a unique triangulation of d 1 n 1. Some alternative interpretations when d n, we can always build any triangulation of d 1 n 1 by locally piecing together triangulations of n n 1 14 / 17
51 Extendability result Theorem (Ceballos-Padrol-S 14) Let d k > n N. Every triangulation of skel k 1 ( d 1 ) n 1 extends to a unique triangulation of d 1 n 1. Some alternative interpretations when d n, we can always build any triangulation of d 1 n 1 by locally piecing together triangulations of n n 1 as d increases, triangulations of d 1 n 1 don t get much more complicated than triangulations of n n 1 14 / 17
52 Extendability result Theorem (Ceballos-Padrol-S 14) Let d k > n N. Every triangulation of skel k 1 ( d 1 ) n 1 extends to a unique triangulation of d 1 n 1. Some alternative interpretations when d n, we can always build any triangulation of d 1 n 1 by locally piecing together triangulations of n n 1 as d increases, triangulations of d 1 n 1 don t get much more complicated than triangulations of n n 1 Moreover, k > n is the best possible. 14 / 17
53 Non-extendability result Theorem (Ceballos-Padrol-S 14) For every n 2 there is a non-extendable triangulation of skeln 1 ( n ) n 1. Idea of proof 15 / 17
54 Non-extendability result Theorem (Ceballos-Padrol-S 14) For every n 2 there is a non-extendable triangulation of skeln 1 ( n ) n 1. Idea of proof 15 / 17
55 Non-extendability result Theorem (Ceballos-Padrol-S 14) For every n 2 there is a non-extendable triangulation of skeln 1 ( n ) n 1. Idea of proof 15 / 17
56 Non-extendability result Theorem (Ceballos-Padrol-S 14) For every n 2 there is a non-extendable triangulation of skeln 1 ( n ) n 1. Idea of proof 15 / 17
57 Non-extendability result Theorem (Ceballos-Padrol-S 14) For every n 2 there is a non-extendable triangulation of skeln 1 ( n ) n 1. Idea of proof 15 / 17
58 Outline of the talk Introduction to triangulations of d 1 n 1 Dyck path triangulations and some relatives Extendability of partial triangulations Further questions 15 / 17
59 Further questions Dyck path triangulations exploit the identity: n C n 1 = ( ) 2n 2, n 1 16 / 17
60 Further questions Dyck path triangulations exploit the identity: n C n 1 = ( ) 2n 2, n 1 Rational Dyck path triangulations use that: ( ) (r + 1)n 2 n C(n, r n 1) =, n 1 16 / 17
61 Further questions Dyck path triangulations exploit the identity: n C n 1 = ( ) 2n 2, n 1 Rational Dyck path triangulations use that: ( ) (r + 1)n 2 n C(n, r n 1) =, n 1 Is there a triangulation involving the following? ( ) ( ) a + b 1 a + b a C(a, b) = or (a+b) C(a, b) = a 1 a where C(a, b) = 1 (for gcd(a, b) = 1) a+b ( a+b ) a are the rational Catalan numbers 16 / 17
62 Take-away messages There is a triangulation of n 1 n 1 involving Dyck paths and cyclic symmetry: If k > n, every triangulation of skel k 1 ( d 1 ) n 1 extends to a unique triangulation of d 1 n 1 For every n 2 there is a non-extendable triangulation of skel n 1 ( n ) n 1 17 / 17
63 17 / 17
64 A tropical translation Theorem (Develin-Sturmfels 04, Santos 04, Ardila-Develin 09) There is a bijection between: triangulations of d 1 n 1 fine mixed subdivisions of d n 1 generic arrangements of d tropical hyperplanes in TP n 1 back 17 / 17
65 A tropical translation Every generic arrangement of d tropical hyperplanes in TP n 1 gives rise to a compatible collection of ( d k) subarrangements: back 17 / 17
66 A tropical translation Every generic arrangement of d tropical hyperplanes in TP n 1 gives rise to a compatible collection of ( d k) subarrangements: back By extendability theorem, the converse holds: if d k > n, every compatible collection of ( d k) subarrangements corresponds to a unique generic arrangement of d tropical hyperplanes in TP n 1 17 / 17
67 A tropical translation Every generic arrangement of d tropical hyperplanes in TP n 1 gives rise to a compatible collection of ( d k) subarrangements: back without adjective tropical, this is Corollary in By extendability theorem, the converse holds: if d k > n, every compatible collection of ( d k) subarrangements corresponds to a unique generic arrangement of d tropical hyperplanes in TP n 1 17 / 17
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